SOLUTION: Evaluate inverse function of f if f(x)={{{x^3+x-1}}}

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Question 1037255: Evaluate inverse function of f if f(x)=x%5E3%2Bx-1
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=x%5E3%2Bx-1

It's a good idea to graph the equation for f(x):

Get some points: (-2,-11), (-1,-3), (0,-1), (1,1), (2,9)



That graph passes both the vertical and horizontal line tests.
Therefore it has an inverse which is a function.

The graph of the inverse function will be the reflection of that 
graph in the green line, which is known as the identity line y=x, 
where y and x are identical.

To find the equation of the inverse.

1. Replace f(x) by y

y=x%5E3%2Bx-1

2.  Interchange x and y

x=y%5E3%2By-1

3. Solve that equation for y.

This involves something pretty advanced.

Write it as

y%5E3%2By-1-x=0

Substitute in Cardano's formula for the solution
to the reduced cubic u%5E3%2Ba%2Au%2Bb=0, which is:



We substitute y for u, 1 for "a" and (-1-x) for "b".



Simplify what we can



Then we change y to f-1(x)



We get some points, using the calculator
 (-11,2), (-10,-1.92), (-7,-1.63), (-3,-1), (-1,0), (1,1), (5,1.63),
(8,1.92), (9,2), (10,2.07)



Edwin